#498422

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(7,8)(9,10,15,21,16,22,13,19)(11,17,14,30,25,20,24,34)(12,26,28,39,29,35,32,33)(18,31,36,23,38,27,37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,14,16,25,13,24)(10,17,21,30,22,20,19,34)(12,27,28,40,29,31,32,23)(18,35,36,33,38,26,37,39);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,23)(13,28)(14,31)(15,32)(16,29)(17,33)(19,36)(20,39)(21,37)(22,38)(24,27)(25,40)(26,34)(30,35);(1,4,5,3)(2,7,6,8)(9,13,16,15)(10,19,22,21)(11,24,25,14)(12,28,29,32)(17,34,20,30)(18,36,38,37)(23,27,40,31)(26,39,35,33)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405618);